The relation of the CTE and CVE is reminiscent of the gravitional anomalies in general: for example, it is known that the non-conservation of the energy-momentum tensor is not independent of the Lorentz anomaly (i.e. antisymmetric \(T^{ab}\) ) and that the anomaly can be shuffled from one to the other, as well as written in several equivalent forms [1]}. The choice where to place the anomaly should be made in terms of the independent degrees of freedom, the Lorentz and torsional anomalies being just manifestations of the presence of degrees of freedom independent from the metric. For hydrodynamic transport, the essential difference is whether the frame fields and connection enter the constitutive relations with sources independently from the metric. In the DC limit both conductivities are equal and constitute two parts: a chemical potential term \(\mu ^2\)  induced by the chiral gauge anomaly and a \(T^2\)  term from the gravitational anomaly. In retrospect, this fits perfectly in to the different ways of understanding torsion in terms of a momentum dependent chiral gauge field corresponding to totally antisymmetric torsion. Finally, in order to discern between the two anomalies one can probe the finite frequency and momentum conductivities, which are different for the CTE and CVE, since the two independent contributions coincide only in the DC limit. On the other hand, it is known that the “anomaly quantization" of the CVE, and therefore also the CTE, is only valid to the lowest non-trivial order \(O(k^2)\)  and that e.g. dynamical gauge fields contribute to the conductivity [2]}.
